23,004 research outputs found
Diffusion Approximations for Online Principal Component Estimation and Global Convergence
In this paper, we propose to adopt the diffusion approximation tools to study
the dynamics of Oja's iteration which is an online stochastic gradient descent
method for the principal component analysis. Oja's iteration maintains a
running estimate of the true principal component from streaming data and enjoys
less temporal and spatial complexities. We show that the Oja's iteration for
the top eigenvector generates a continuous-state discrete-time Markov chain
over the unit sphere. We characterize the Oja's iteration in three phases using
diffusion approximation and weak convergence tools. Our three-phase analysis
further provides a finite-sample error bound for the running estimate, which
matches the minimax information lower bound for principal component analysis
under the additional assumption of bounded samples.Comment: Appeared in NIPS 201
A note on Probably Certifiably Correct algorithms
Many optimization problems of interest are known to be intractable, and while
there are often heuristics that are known to work on typical instances, it is
usually not easy to determine a posteriori whether the optimal solution was
found. In this short note, we discuss algorithms that not only solve the
problem on typical instances, but also provide a posteriori certificates of
optimality, probably certifiably correct (PCC) algorithms. As an illustrative
example, we present a fast PCC algorithm for minimum bisection under the
stochastic block model and briefly discuss other examples
DROP: Dimensionality Reduction Optimization for Time Series
Dimensionality reduction is a critical step in scaling machine learning
pipelines. Principal component analysis (PCA) is a standard tool for
dimensionality reduction, but performing PCA over a full dataset can be
prohibitively expensive. As a result, theoretical work has studied the
effectiveness of iterative, stochastic PCA methods that operate over data
samples. However, termination conditions for stochastic PCA either execute for
a predetermined number of iterations, or until convergence of the solution,
frequently sampling too many or too few datapoints for end-to-end runtime
improvements. We show how accounting for downstream analytics operations during
DR via PCA allows stochastic methods to efficiently terminate after operating
over small (e.g., 1%) subsamples of input data, reducing whole workload
runtime. Leveraging this, we propose DROP, a DR optimizer that enables speedups
of up to 5x over Singular-Value-Decomposition-based PCA techniques, and exceeds
conventional approaches like FFT and PAA by up to 16x in end-to-end workloads
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